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Exact structures for persistence modules

Benjamin Blanchette, Thomas Brüstle, Eric J. Hanson

TL;DR

This paper develops a rigorous framework for invariants of multiparameter persistence modules using exact structures and relative homological algebra. It establishes the theory of exact structures, relative projectives, and homological invariants, linking them to dimension and rank invariants, and explores global- and representation-dimension considerations, especially for finite grids. A key contribution is an adjunction-based extension-contraction machinery that lifts finite-poset results to certain infinite posets, enabling controlled transfer of exact-structure results via finite aligned subgrids. The work also provides explicit descriptions of irreducible morphisms between relative projectives for several exact structures, enabling computation of endomorphism-quivers and stability analyses. Overall, the paper deepens the toolkit for invariants in multiparameter persistence and clarifies when stability and finiteness results can be expected, as well as their limitations in infinite settings.

Abstract

We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, classical arguments about the relative projective modules of an exact structure make use of Auslander-Reiten theory. One of our results establishes a new adjunction which allows us to ``lift'' these arguments to certain infinite posets over which Auslander-Reiten sequences do not always exist. We give several examples of this lifting, in particular highlighting the non-existence and existence of resolutions by upsets when working with finitely presentable representations of the plane and of the closure of the positive quadrant, respectively. We then restrict our attention to finite posets. In this setting, we discuss the relationship between the global dimension of an exact structure and the representation dimension of the incidence algebra of the poset. We conclude with our second novel contribution. This is an explicit description of the irreducible morphisms between relative projective modules for several exact structures which have appeared previously in the literature.

Exact structures for persistence modules

TL;DR

This paper develops a rigorous framework for invariants of multiparameter persistence modules using exact structures and relative homological algebra. It establishes the theory of exact structures, relative projectives, and homological invariants, linking them to dimension and rank invariants, and explores global- and representation-dimension considerations, especially for finite grids. A key contribution is an adjunction-based extension-contraction machinery that lifts finite-poset results to certain infinite posets, enabling controlled transfer of exact-structure results via finite aligned subgrids. The work also provides explicit descriptions of irreducible morphisms between relative projectives for several exact structures, enabling computation of endomorphism-quivers and stability analyses. Overall, the paper deepens the toolkit for invariants in multiparameter persistence and clarifies when stability and finiteness results can be expected, as well as their limitations in infinite settings.

Abstract

We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, classical arguments about the relative projective modules of an exact structure make use of Auslander-Reiten theory. One of our results establishes a new adjunction which allows us to ``lift'' these arguments to certain infinite posets over which Auslander-Reiten sequences do not always exist. We give several examples of this lifting, in particular highlighting the non-existence and existence of resolutions by upsets when working with finitely presentable representations of the plane and of the closure of the positive quadrant, respectively. We then restrict our attention to finite posets. In this setting, we discuss the relationship between the global dimension of an exact structure and the representation dimension of the incidence algebra of the poset. We conclude with our second novel contribution. This is an explicit description of the irreducible morphisms between relative projective modules for several exact structures which have appeared previously in the literature.
Paper Structure (17 sections, 29 theorems, 19 equations, 2 figures)

This paper contains 17 sections, 29 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Posets and lattices
  4. Persistence modules
  5. Invariants
  6. General theory of exact structures
  7. Exact structures
  8. Relative projective objects
  9. Homological invariants
  10. Approximations and resolutions
  11. Homological invariants
  12. Global dimensions and the representation dimension
  13. Characterizations of irreducible morphisms
  14. Exact structures for infinite posets
  15. The restriction and extension functors
  16. ...and 2 more sections

Key Result

Proposition 2.7

Let $\mathcal{S},\mathcal{T} \subseteq \mathcal{P}$ be spreads. We denote by $\mathfrak{I}(\mathcal{S},\mathcal{T})$ the set of connected components $\mathcal{U}$ of $\mathcal{S} \cap \mathcal{T}$ which satisfy Then $\dim_\mathbb{K}\mathop{\mathrm{\mathrm{Hom}}}\nolimits_\mathcal{P}\left(\mathbb{I}_{\mathcal{S}},\mathbb{I}_{\mathcal{T}}\right) = |\mathfrak{I}(\mathcal{S},\mathcal{T})|$. Moreover,

Figures (2)

  • Figure 1: (i) The closed upset $\langle \{a_1,a_2,a_3\},\infty\langle$. This can also be written as $\rangle \{c_1,c_2\},d\rangle$. (ii) The segment $\langle a,b\rangle$, the closed upset $\langle a,\infty\rangle$, and the cohook $\rangle c,b\rangle$ all coincide. (iii) The hook $\langle a,b\langle$. All examples are taken over the poset $\{0,1,2\}\times \{0,1,2\}$ with the usual product order.
  • Figure 2: The module $M = \mathbb{I}_{\langle \{(01),(10)\},(21)\langle} \in \mathop{\mathrm{\mathrm{rep}}}\nolimits(\mathbb{R}^2)$ from Example \ref{['ex:contraction']}.

Theorems & Definitions (109)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Remark 2.6
  • Proposition 2.7
  • Remark 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 99 more